I've seen transparrent paper that looks like polyethylene. In fact, it's a rolling paper. It burns like paper, its smoke smells like paper and it is only slightly sweet to taste but is water proof, not getting moist.
Now, this thing, I believe, is a natural cellulose, treated to order the fibers so that it gets an ordered "crystal" structure.
How it is done, I have no idea, but I think that it was known for hundreds of years because russians were using it for wrapping flowers in flower shops etc.
So... I assume that this polymer is made from cellulose short subunits by some technique.